Question: Simplify the following expression and state the condition under which the simplification is valid: $k = \dfrac{x^2 + 10x}{x^2 + 13x + 30}$
First factor the expressions in the numerator and denominator. $ \dfrac{x^2 + 10x}{x^2 + 13x + 30} = \dfrac{(x)(x + 10)}{(x + 3)(x + 10)} $ Notice that the term $(x + 10)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(x + 10)$ gives: $k = \dfrac{x}{x + 3}$ Since we divided by $(x + 10)$, $x \neq -10$. $k = \dfrac{x}{x + 3}; \space x \neq -10$